A lateral edge of the hexagonal pyramid is 2.0m long and its base perimeter is 6m determine the volume of the pyramid and its lateral area
The base perimeter is 6 m, so each side of the hexagon is 1 m
Let's concentrate on one of the triangles of the pyramid, it will be isosceles with sides
2 m, 2m and 1 m. Draw a perpendicular and you will have a right-angled triangle, let
the height be h
then h^2 + (1/2)^2 = 2^2
h^2 = 15/4
h = √15/2
So the area of one of those triangles = (1/2)(1)(√15/2) = √15/4
So 6 of them would be 3√15/2 m^2 <---- lateral surface area
No look at the base, it is also made up of 6 isosceles triangle, sketch one of them
the area of one of them = (1/2)(1)(1)sin60° = (1/2)(√3/2) = √3/4
and with 6 of those, we get a base area of 3√3/2 m^2
We need the height of the pyramid, call it k
Back to the equilateral triangle of the base, all sides have 1, so with basic
geometry we see that the "height" in one of these is √3/2
k^2 + (√3/2)^2 = (√15/2)^2
k^2 + 3/4 = 15/4
k^2 = 3
k = √3, <---- nice
volume of pyramid = (1/3)(base area)(height)
= (1/3)(3√3/2)(√3) = 3/2 m^3
better check my arithmetic on that, should have written it out first.
To determine the volume and lateral area of the hexagonal pyramid, we'll need some additional information. Specifically, we need either the apothem (distance from the center of the base to the mid-point of a side) or the height of the pyramid.
Let's consider both cases:
Case 1: The apothem is known:
If the apothem is given, we can use the formula for the volume and lateral area of a regular hexagonal pyramid.
The volume (V) of a hexagonal pyramid is given by:
V = (1/3) × Base Area × Height
The base area of a regular hexagon can be calculated using the formula:
Base Area = (3√3 × side^2)/2
The lateral area (LA) of a hexagonal pyramid is given by:
LA = Perimeter of Base × Slant Height / 2
Since the perimeter of the base is given as 6m, we can divide it by 6 to find the side length of the hexagon.
Case 2: The height is known:
If the height is given, we can use a different formula to calculate the volume and lateral area of the hexagonal pyramid.
The volume (V) of any pyramid is given by:
V = (1/3) × Base Area × Height
The base area of a regular hexagon can be calculated using the formula:
Base Area = (3√3 × side^2)/2
The lateral area (LA) of a pyramid is given by:
LA = (Perimeter of Base × Slant Height)/2
In this case, we need to find the slant height, which can be calculated using the height and the side length.
Once we have either the apothem or the height, we can substitute the values into the formulas to find the volume and lateral area of the hexagonal pyramid.